In pentagon ABCDE , BC = CD = DE = 2 units, angle E is a right angle and angle B = angle C = angle D = 60 . The length of segment can be expressed in simplest radical form as AE = a + b*sqrt(c) units. What is the value of a + b + c?

Guest Sep 5, 2021

#1**+1 **

iirc, this is very similar to a previous mathcounts nationals problem. here is the solution to that from my document

CFD is a 90, 45, 45 triangle, so CF and DF is $\sqrt2$

$BE = \sqrt(2(2+\sqrt(2))^2) = \sqrt{2} \cdot (2 + \sqrt{2})= 2\sqrt{2} + 2$

$AE^2 = AB^2 + BE^2 = 2 \times (2\sqrt{2} + 2)^2$

$AE = \sqrt{2} \times (2\sqrt{2} + 2) = 4 + 2\sqrt{2}$

$a + b + c = 2 + 4 + 2= \boxed{8}$

HighSchoolDx Sep 5, 2021